In this thesis, we consider a linear second order ordinary differential equation (ode)

In this thesis, we consider a linear second order ordinary differential equation (ODE),

where is a continuous function, , and are given constants, and . We are interested to find a particular solution for Equ. (1).
Applications of Second Order Ordinary Differential Equations
The socond order ordinary differential equation (1) can model many different phenomena that we encounter every day. For example, the equation

represents what happens when an object is subject to a force towards an equilibrium position with the magnitude of the force being proportional to the distance from equilibrium. The above equation can be considered as an approximation to the equation of motion of a particular point on the basilar membrane, or anywhere else along the chain of transmission between the outside air and the cochles [10]. Thus the solution of the differential equation can help explain the perception of pitch and intensity of musical instruments by human ears.
We may also use a second order ODE to model at what sltitude a skydiver's parachute must open before he reaches the ground so that he lands safely. To model this phenomenon, we let denote the altitude of the skydiver. We consider Newton's second law

where represents the force, represents the mass of the skydiver and represents the acceleration. We assume that the only forces acting on the skydiver are air resistance and gravity when the skydiver falls through the air toward the earth. We also assume that the air resistance is proportional to the speed of the skydiver with being the positive constant of proportionslity known as the damping constant. The Newton's second law (2) then translates to

If we know the initial height and velocity at the time of jump, we have an initial value problem which is the equation (3) together with the initial conditions,